(2/5)^x+3=(125/8)^x-1.(0 4)^2x-3

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Jun 10, 2024

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Solving the Equation: (2/5)^x + 3 = (125/8)^x - 1 - (0.4)^2x - 3

In this article, we will solve the equation (2/5)^x + 3 = (125/8)^x - 1 - (0.4)^2x - 3. This equation involves exponential functions with different bases and exponents, making it a challenging problem. Let's break it down step by step and find the solution.

Step 1: Simplify the Equation

First, let's simplify the equation by rewriting the fractions with a common denominator:

(2/5)^x + 3 = (125/8)^x - 1 - (2/5)^2x - 3

Now, let's rewrite the equation in a more compact form:

(2/5)^x + (2/5)^2x + 4 = (125/8)^x - 1

Step 2: Identify the Pattern

Observe that the left-hand side of the equation has two terms with the same base (2/5), with exponents x and 2x. We can rewrite the equation as:

(2/5)^x (1 + (2/5)^x) + 4 = (125/8)^x - 1

Step 3: Find the Solution

Now, we can solve for x. Let's start by equating the two exponential functions:

(2/5)^x (1 + (2/5)^x) = (125/8)^x - 5

Notice that (125/8) = (5/2)^3, so we can rewrite the equation as:

(2/5)^x (1 + (2/5)^x) = (5/2)^(3x) - 5

Step 4: Solve for x

To solve for x, we can use the property of exponential functions that states:

a^x = b^x => x = log_a(b)

Applying this property, we get:

x = log_(2/5) ((5/2)^(3x) - 5)

Solving for x algebraically, we get:

x = log_(2/5) ((5/2)^3 - 5)

Conclusion

In this article, we solved the equation (2/5)^x + 3 = (125/8)^x - 1 - (0.4)^2x - 3 by simplifying the equation, identifying the pattern, and using the properties of exponential functions. The solution is x = log_(2/5) ((5/2)^3 - 5).